Inferring Option Prices and Volatility from Spread Quotes

# Calculate options prices based on given options and spread prices


# Calculate options prices based on given options and spread prices












Suppose you know the following information:

- Futures price on a stock is 66

- 70 strike straddle is trading at 27

- 50-60 put spread is trading at 2.5

- 50-60-70 put butterfly is trading at 0.2

- Assume volatility is constant across strikes; interest rate is 0

Questions:

- What are the fair values for the 80-strike call, 60-strike straddle, and 40-strike put

- Now assume we have a volatility smile among the curve, how would this change your markets differently

My try:

Using put-call parity and direct definitions of the spreads, I have below equations

Call(K=70) - Put(K=70) = (Futures - K) = (66-70)

Call(K=70) + Put(K=70) = 27

Put(K=60) - Put(K=50) = 2.5

Put(K=50) + Put(K=70) - 2Put(K=60) = 0.2

Solving the above equations, I got:

Call(K=70) = 11.5

Put(K=70) = 15.5

Put(K=50) = 10.7

Put(K=60) = 13.2

Given the assumption of constant volatility, I am not sure how I should go from here to calculate values for:

Call(K=80)

Call(K=60) + Put(K=60)

Put(K=40)

Any help or hint is highly appreciated!

## Answer by Kermittfrog (score 1)

https://quant.stackexchange.com/a/68791

What you are given is a linear combination in instruments and corresponding (benchmark) prices, what you need are

- invert the linear combinations to arrive at the benchmark prices

- back out implied vols from benchmark prices

- apply the vols to your new products.

For the first step, I'd go as

$$ \begin{align} Ax&=b\\ \begin{pmatrix}1&0&0&0&0\\ 0 & 0 & 0 & 1 & 1\\ 1 & 0 & 0 & 1 & -1\\ 0&1&-1&0&0\\ 0 & 1 & -2 & 1 & 0 \end{pmatrix}\begin{pmatrix}F\\P(50)\\P(60)\\P(70)\\C(70)\end{pmatrix}&=\begin{pmatrix}66\\27\\70\\2.5\\0.2\end{pmatrix} \\ \Rightarrow \quad x&=A^{-1}b\\ &=\begin{pmatrix}66\\20.3\\17.8\\15.5\\11.5\end{pmatrix} \end{align} $$

We can now back out the implied total variance (assuming zero interest rate) using the Black-Scholes-Model:

$$ \begin{align} C(\sigma_X^2T)&\equiv F\mathrm{N}\left(\frac{\ln(F)-ln(X)+\frac{1}{2}\sigma_X^2T}{\sigma_X\sqrt{T}} \right)-X\mathrm{N}\left(\frac{\ln(F)-ln(X)-\frac{1}{2}\sigma_X^2T}{\sigma_X\sqrt{T}} \right)\\ &\stackrel{!}{=}O(X) \end{align} $$ (accordingly for puts) where $\sigma_X^2T$, the total variance, is unknown and $O(X)$, the observed price, is given.

Using some root search method, you can now calculate the implied vols across all given strikes and option prices and obtain $\sigma_{50}^2T\approx 1.64192027$, $\sigma_{60}^2T\approx 0.71805172$, $\sigma_{70}^2T\approx 0.2494285$. You can then use these vols to price your other products.

NB: Question 1 may be ill-defined under the assumption of constant vols; probably you just have to 'pick' one vol? Question 2 then needs to be answered using the vols we just backed out.

HTH?